C₃×#90527 #13061178

order	1344
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#90527

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₇ ⋊ #577
Sylow	C₇ ⋊ C₃×#577

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₇
Fitting	C₃×C₇×H(5,𝔽₂)	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₃×H(5,𝔽₂)	D₇
hypocenter	C₇	C₃×#577
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ C₃×H(5,𝔽₂)

516 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 214 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2,3)(4,5,7,9,10,8,6)(11,15,17,14)(12,20,21,26)(13,22,16,23)(18,19,25,24);(11,14,17,15)(12,19,21,24)(13,22,16,23)(18,20,25,26);(11,13,17,16)(12,18,21,25)(14,23,15,22)(19,26,24,20);(5,6)(7,8)(9,10)(11,12)(13,18)(14,19)(15,24)(16,25)(17,21)(20,22)(23,26)

Conjugacy classes & character table

219 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 10467);